Defining the Abatement Cost in Presence of Learning-by-Doing: Application to the Fuel Cell Electric Vehicle

Abstract

We consider a partial equilibrium model to study the optimal phasing out of polluting goods by green goods. The unit production cost of the green goods involves convexity and learning-by-doing. The total cost for the social planner includes the private cost of production and the social cost of carbon, assumed to be exogenous and growing at the social discount rate. Under these assumptions the optimization problem can be decomposed in two questions: (i) when to launch a given schedule; (ii) at which rate the transition should be completed that is, the design of a transition schedule as such. The first question can be solved using a simple indicator interpreted as the MAC of the whole schedule, possibly non optimal. The case of hydrogen vehicle (Fuel Cell Electric Vehicles) offers an illustration of our results. Using data from the German market we show that the 2015–2050 trajectory foreseen by the industry would be consistent with a carbon price at 52€/t. The transition cost to achieve a 7.5 M car park in 2050 is estimated at 21.6 billion € that is, to JEl 4% discount rate, 115 € annually for each vehicle which would abate 2.18 tCO\(_2\) per year.

JEL Classification

Appendix A: proof of Lemma 1

Proof

To minimize the total cost (2), let us introduce \(\lambda _t\) the co-state variable associated to the relation (3), and \(\theta _t\) and \(\delta _t\) the Lagrange multipliers associated to the two constraints (4) on \(x_t\): it is positive (\(\theta _t\)) and smaller that the total fleet size (\(\delta _t\)). The first order conditions (together with the complementarity slackness conditions) are:

The main step of the proof consists in proving that \(x_{t}\) is increasing if \(x_{t}\in (0,N)\). This condition ensures that once \(x_{t}>0\) the number of green cars cannot come back to zero, and that \(x_{t}\) does not move when \(x_{t}=N\). If \(x_{t}\) is strictly positive ( \(\theta _{t}=0\)) and lower than the total car fleet (\(\delta _{t}=0\)), Eq. (19) becomes \( C_{x}(X_{t},x_{t})-c_{o} =p_{t}^{CO2}+\lambda _{t} \) and taking the time derivative:

The last term of the right hand side is positive because \(C_{X}(X,x)\) is concave with respect to x and \(C_{X}(X,0)=0\) (since \(C(X,0)=0, \forall X\)). Since \(C_{xx}\), \( p^{CO2}_t\) and r are all positive, \(\dot{x}\) is also positive so that \(x_t\) is increasing through time.

Then, since the CO\(_{2}\) price increases exponentially, \(x_{t}\) cannot be always null along an optimal trajectory. Then either \(x_{0}=0\) or \(x_{0}>0\). In the latter case \(T_{s}=0\), whereas in the former case \(T_{s}\) is the inf of the dates at which \(x_{t}>0\).

The ending date is finite, \(T_e<+\infty \): From the above proof, when \(x_t\) is positive its time derivative is bounded below by a positive number, so \(x_t\) necessarily reaches N in a finite time.

Proof

If \(C_{xx}=0\), given that \(C(X,0)=0\), \(C_{X}(X,x)=C_{Xx}(X,x)x\). Then, we resort by reductio ad absurdum assuming \(T_{s}<T_{e}\). Between the two dates the Eq. (5) is satisfied and taking its derivative with respect to t gives:

Along the optimal trajectory, \(T_{s}\) should minimize this function. Taking the derivative with respect to \(T_{s}\) in the equation above and setting it equal to zero gives the Eq. (6). \(\square \)

From Lemma 1, the optimal trajectory \((x_t^*)_{t\in [0,+\infty )}\) can be described as the launching of a deployment schedule. There is therefore no loss to minimize the cost over the set of trajectories defined as the launching (and ending) of a deployment schedule. The trajectory obtained from the optimal launching of the optimal deployment schedule coincides with the optimal trajectory described in Lemma 3.

Proof

Using the decomposition of the total discounted cost \(\Gamma \) provided by Eq. (11), the schedule \(\xi \) only influences the DAC and no other component of the cost. So that the \(\xi \) that minimizes \(\Gamma \) corresponds to the \(\xi \) that minimizes the DAC. A deployment schedule \((\bar{X},D,\xi )\) and its DAC are only defined for \(\xi \) such that \(\int _0^D \xi _\tau d\tau =\bar{X}\). The optimal \(\xi \), for a given \(\bar{X}\) and D, is then the solution of the optimization program:

Together with the optimality conditions satisfied by \(\bar{X}^*\), \(D^*\) and \(T_l^*\), to be studied below, these first order conditions will coincide with 5. However, even for suboptimal \(\bar{X}\) and D the schedule \(\xi \) should satisfies these equations, which then gives the minimized deployment cost \(I^*(\bar{X},D)\) and the associated \(DAC(\bar{X},D)\). The derivative of the deployment cost are:

There are two possible strategies to prove that \(\bar{X}^*\) and \(D^*\) are independent of the CO\(_2\) price \(p_0\). One is sketched in the main text. The other consists in looking at first order conditions and showing that the optimal duration and accumulated quantity satisfy a pair of equation independent from the CO\(_2\) price.

Proof

From the expression (11) of the total discounted cost, the optimal \(D^*\), \(\bar{X}^*\) and launching date satisfy the equations:

The first equation corresponds to Eq. (13), the third correspond to (15). And injecting the third into the second gives Eq. (14) satisfied by \(\bar{X}^*\).

The two Eqs. (13) and (14) are independent of the CO\(_2\) price or the launching date, the couple \(\bar{X}^*\) and \(D^*\) is therefore independent of \(p_0\), and so is the optimal deployment schedule \((\xi _\tau ^*)_{\tau \in [0,D]}\) associated to them.

In addition, it is interesting to see how these equations together with Eq. (23) give back the Eq. (5).

which is similar to (8) except that the second line above, the value of interim abatement, replaces \(p_0\bar{X}\). The derivative of the second line with respect to \(T_l\) is (after an integration by parts):

IPCC (2013) Climate Change 2013: the physical science basis. contribution of working group I to the fifth assessment report of the intergovernmental panel on climate change, Cambridge University Press, Cambridge, United Kingdom and New York, USA, chapter Summary for PolicymakersGoogle Scholar